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Bezout's Theorem

Bezout was a famous mathematician who discovered many beautiful formulas. One of his most famous theorem/identities dealt with numbers and their greatest common factors. His theorem states, if integers \(a\) and \(b\) are relatively prime, then there exists \(x\) and \(y\), integers to satisfy the equation \(ax+by=1\). For any integers \(a\) and \(b\) excluding zero, let \(d\) be the greatest common divisor such that \(d\) equals the gcd of \(a\) and \(b\) in other words, \(d\)\(=\)gcd\(\left(a,b\right)\). If this is true then there must exist integers known as \(x\) and \(y\) to satisfy the equation \(ax+by=d\).

There are many important facts that go along with Bezout’s identity:

1.)    All common divisors of \(d\) are common divisors of \(a\) and \(b\) as well
2.)    As for the fact above, all divisors of \(a\) and \(b\) are also divisors of \(d\)
3.)    \(a\)/\(d\) and \(b\)/\(d\) are prime integers
4.)    \(a\)/\(d\)\(\left(x\right)\)+\(b\)/\(d\)\(\left(y\right)\)\(=1\)
5.)    The greatest common divisor \(d\) is actually the smallest integer that can be written to satisy \(ax+by\)
6.)    All integers of the form \(ax+by\) are multiples of \(d\)

Next we will work out an example.

If we have \(2x+3y=1\), well by trial and error we can quickly see that \(x\) is \(2\) while \(y\) is \(-1\). But when we are given huge numbers it can be a bit more difficult. In order to work the equation out with nigger numbers the Euclidean algorithm must be applied. Let’s recall that Euclid’s theorem states: For any numbers and \(a,b,\) \(x,\) gcd \(\left(a,b\right)\)= gcd\(\left(b,a-bx\right)\)